8.8: Kohn-Sham functional

For solids, we have $10^{26}$ electron states. Analytic solution becomes impossible. In the past 20 years the density functional theory has come to dominate condensed matter physics, extending to chemistry, materials, minerals and beyond.

A popular form of DFT functional was introduced by Nobel laureate Walter Kohn and Lu Sham:

$$F(\rho ) = T[\rho ] + \frac{1}{2} \int \frac{\rho ( {\bf r})\rho ({\bf r}')}{4\pi \epsilon_0 |{\bf r − r}' |} d^3 {\bf r}d^3 {\bf r}' + E_{xc}[\rho ] + \sum_i \int \frac{Z_i e\rho ({\bf r}')}{4\pi \epsilon_0|{\bf R_i − r}' |} d^3 {\bf r}' \nonumber$$

Nobody has found a satisfactory functional for $T$. What is generally used is:

$$− \frac{\hbar^2}{2m} \sum_i \int \phi_i \nabla^2_i \phi_i d^3 {\bf r} \nonumber$$

which is the kinetic energy of non-interacting “quasiparticles” and depends explicitly on the wavefunctions. The integrals represent electrostatic interactions between the electrons and between electrons and ions, and $E_{xc}$ is ‘everything else’. The advantage of this form is that it can be recast to give a set of one-particle equations with non-interacting fermions moving in an effective potential:

$$V_{ef f} = \sum_{ion} \frac{Ze}{4\pi \epsilon_0|{\bf R_{ion} − r}'|} + \int \frac{\rho ({\bf r}')}{4\pi \epsilon_0|{\bf r − r}' |} d^3 {\bf r} + \frac{\delta E_{xc}[\rho ]}{\delta \rho ({\bf r})} \nonumber$$

Since $V_{ef f}$ depends on $\rho ({\bf r})$ these equations must be solved self-consistently.

Thus the density functional theorem shows that the problem of solving the Schrödinger equation for a collection of interacting electrons can be transformed to that of a system of non-interacting ‘quasiparticles’, with the cost that the Hamiltonian depends on the electron density $\rho ({\bf r})$:

$$H[\rho ({\bf r})]\phi_i = E_i\phi_i \quad \text{ where } \quad \rho ({\bf r}) = \sum_i |\phi_i({\bf r})|^2 \nonumber$$

Thus the Schrödinger equation is a nonlinear differential equation of many variables. Thus we must turn to the variational method. The most general approach here is to use a Fourier Series (plane wave basis set). The wavefunction for the $i$th electron is then written as

$$\phi_i = \sum_k c_{ik} \text{ exp}(−i{\bf k.r}) \quad \text{ and the variational equation becomes: } \quad E_0 = \text{Min} \sum_i \langle \phi_i |\hat{H} (\rho )|\phi_i \rangle \nonumber$$

The accuracy of the ground state energy of the electrons is determined by the number of Fourier components used. The wavefunctions are expanded in a computer-friendly basis set and the variational principle is used to transform the problem from a set coupled non-linear differential equations into a minimisation of a single function of many variables. Most structural properties of materials depend only on the electron ground state.

The single particle eigenstates of Kohn-Sham functional are not proper single electron states: indistinguishability means there is no such thing. Nevertheless, they are Bloch states, and they do exhibit well defined symmetry and energy “band-structure” which can help with interpretation of the electronic structure